2.4.1 Quality vs. other independent variables

And now we plot the rest of the variables against the quality:

2.4.2 Additional plots

In this section, we plot other graphs to see how the data is distributed based on the correlation matrix

Faulty wine

Normal wine

Good wine

4.1.1 Simple multinomial logistic regression

The first test we are going to do is the multinomial logistic regression. The idea is simple, we will try to derive the taste quality parameter based on the other 12 independent variables. A summary of the regression is shown below

## Call:
## multinom(formula = taste ~ fixed.acidity + volatile.acidity + 
##     citric.acid + residual.sugar + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + density + pH + sulphates + alcohol, 
##     data = training_redwine)
## 
## Coefficients:
##        (Intercept) fixed.acidity volatile.acidity citric.acid
## faulty    405.6065     0.6609973         4.697646   0.9022834
## good      165.1007     0.2172616        -2.580776   0.5805793
##        residual.sugar chlorides free.sulfur.dioxide total.sulfur.dioxide
## faulty      0.3739949  7.039428        -0.025045965          -0.01196597
## good        0.2213255 -8.453456         0.008314928          -0.01831228
##          density        pH sulphates    alcohol
## faulty -435.8021 7.1278877 -1.142488 -0.6364384
## good   -179.8604 0.4315777  3.755149  0.7501815
## 
## Std. Errors:
##        (Intercept) fixed.acidity volatile.acidity citric.acid
## faulty    3.043451    0.16482994        0.9103453   1.3053491
## good      1.871254    0.08843828        0.8479439   0.9097131
##        residual.sugar chlorides free.sulfur.dioxide total.sulfur.dioxide
## faulty     0.08221751  3.288558          0.02391135          0.008781613
## good       0.06611148  3.531090          0.01358676          0.005692609
##         density        pH sulphates    alcohol
## faulty 2.967102 1.5336428 1.3665900 0.18444246
## good   1.825386 0.9346179 0.5627972 0.09671016
## 
## Residual Deviance: 1073.626 
## AIC: 1121.626

As multinomial logistic regression does not provide the p-values, we will calculate them by normalizing the results. The calculated p-values are as follows:

The model summary output has a block of coefficients and a block of standard errors. Each of these blocks has one row of values corresponding to a model equation. Focusing on the block of coefficients, we can look at the first row comparing prog = “faulty” to our baseline prog = “average” and the second row comparing prog = “good” to our baseline prog = “average”. If we consider our coefficients from the first row to be b1 and our coefficients from the second row to be b2, we can write our model equations as follows

\[\ln(\dfrac{P(prog=faulty)}{P(prob=average)})=b_{10}+b_{11}+b_{12}+b_{13}\] \[\ln(\dfrac{P(prog=good)}{P(prob=average)})=b_{20}+b_{21}+b_{22}+b_{23}\] We can also use predicted probabilities to help you understand the model. We can calculate predicted probabilities for each of our outcome levels using the fitted function. We can start by generating the predicted probabilities for the observations in our dataset and viewing the first few rows

We can predict the test values based on this regression:

Based on the outputs, we have an missclassification error of 14 %. Also, we can plot the ROC curve. The ROC curve illustrates the performance of a binary classifier system as its discrimination threshold varies. The curve shows the true positive rate against the false positive rate at various threshold settings. The true-positive rate is also known as recall and the false-positive rate is also known as the fall-out or probability of false alarm.

The area under the curve is 88.0051764%.

Note: Display AUC value: 90+% - excellent, 80-90% - very good, 70-80% - good, 60-70% - so so, below 60% - not much value.

4.1.2 Stepwise multinomial logistic regression

In this case, we use stepwise regression to find the best parameters. The summary of the simulation is as follows

Again, we need to calculate the new p-values based on the previous results

We can predict the test values based on the stepwise multimodal regression:

Based on the outputs, we have an missclassification error of 14.6666667 %. Also, we can plot the ROC curve. The ROC curve illustrates the performance of a binary classifier system as its discrimination threshold varies. The curve shows the true positive rate against the false positive rate at various threshold settings. The true-positive rate is also known as recall and the false-positive rate is also known as the fall-out or probability of false alarm.

The area under the curve is 87.9951056%.

Note: Display AUC value: 90+% - excellent, 80-90% - very good, 70-80% - good, 60-70% - so so, below 60% - not much value.

4.1.3 Summary of the results

With the simple multinomial regression taking into account all the variables, we are able to predict the quality of wine with a missclassification error of 14 %. Based on the p-values, the parameters that characterize a good wine are as follows:

• fixed acidity: Increase fixed acidity
• Volatile acidity: Decrese volatile acidity
• Residual Sugar: Increase residual sugar
• Density: Reduce the density
• Chlorides: Reduce cholrides
• Total SO2: Reduce total SO2
• Sulphates: Increase the sulphates
• Alcohol: Increase the alcohol

Also, based on the results, the key differences of a good wine from a bad wine are found in the Total SO2, Sulphates and the Alcohol level. We will need topay attention to those values.

4.2.1 Refining the parameters

It is time now to run a classification algorithm on the data set. We have chosen to use the random forest tree algorithm for this.

First of all, we have a look at the confusion matrix results:

And now the same confusion matrix from a percentage in class perspective (rows sum too 100%)

This is how the error looks like:

After several trials, it seems that the error tends toget stabilized after the 80 trees. We have selected 128 trees.

For the final version of the model, we tried several combinations until we found the ones we like:

4.2.2 Prediction based on the parameters

After running the predictionw with the testing sample, the following confusion matrix is obtained:

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction average faulty good
##    average     109      2    2
##    faulty       64      3    0
##    good         80      2   38
## 
## Overall Statistics
##                                         
##                Accuracy : 0.5           
##                  95% CI : (0.442, 0.558)
##     No Information Rate : 0.8433        
##     P-Value [Acc > NIR] : 1             
##                                         
##                   Kappa : 0.1985        
##                                         
##  Mcnemar's Test P-Value : <2e-16        
## 
## Statistics by Class:
## 
##                      Class: average Class: faulty Class: good
## Sensitivity                  0.4308       0.42857      0.9500
## Specificity                  0.9149       0.78157      0.6846
## Pos Pred Value               0.9646       0.04478      0.3167
## Neg Pred Value               0.2299       0.98283      0.9889
## Prevalence                   0.8433       0.02333      0.1333
## Detection Rate               0.3633       0.01000      0.1267
## Detection Prevalence         0.3767       0.22333      0.4000
## Balanced Accuracy            0.6729       0.60507      0.8173

The ROC curve illustrates is as follows.

The area under the curve is 94.270975%.

Note: Display AUC value: 90+% - excellent, 80-90% - very good, 70-80% - good, 60-70% - so so, below 60% - not much value.

4.2.3 Summary of the random forest

The model is strongly biased towards the faulty classification. The model is pretty good at predicting the good wines, and this is exacly what we are looking for. Although the model might seem to be terrible at dealing with the average wines (usually splitting them 47% average, 24% faulty, 28% good) we are not concerned with upward misclassifications. Also, with such an abundance of average wines, losing out on 47% of them is not so terrible. The main objective is to select the best wines to increase our revenues.

The variable importance is as follows:

Based on the importance, the parameters that characterize a good wine are as follows:

• Alcohol: Increase the alcohol
• Total SO2: Reduce total SO2
• Sulphates: Increase the sulphates
• Residual Sugar: Increase residual sugar
• Density: Reduce the density
• Volatile acidity: Decrese volatile acidity